Who understands math well enough to teach it to third graders?

ANN ARBOR, Mich.—More than 15 years of research shows most Americans and most teachers lack sound mathematical skills, leaving U.S. 12th grade math students trailing their peers in 21 other nations.

Many teachers lack what a group of University of Michigan researchers calls mathematical knowledge for teaching. Knowing mathematics for teaching is more than the ability to solve a problem or to tell a pupil to just move the decimal point over.

Effective mathematics teaching also requires knowing why a procedure works, being able to interpret what led to a student error, or choosing a model that accurately shows an idea, the researchers say.

"If we argue that there is professional knowledge for teaching mathematics, then we have to show that improving this knowledge also enhances student achievement," said Deborah Lowenberg Ball, dean of the U-M School of Education, who authored the article with colleagues Heather Hill and Hyman Bass.

The better that teachers are at understanding mathematics in ways that give them flexibility and insight, the more their students learn, the researchers argue in the fall issue of the journal American Educator, drawing on work developed at U-M over eight years.

Ball, who helped organize the Center for Proficiency in Teaching Mathematics, has spent more than 20 years investigating specialized mathematical skills that teachers need. She notes that it's not the same set of math skills required to be a successful accountant, carpenter or engineer.

The teacher must not only know how to solve problems but also know how to translate that knowledge into the language of a small child, and how to understand the logic of students who use unconventional self-invented methods that might work to solve some problems but not others, she says.

A teacher needs to be able to notice subtle shifts in what children say that may reflect deeper misunderstandings. For example, when Ball taught third grade, she found that her students often referred to the number zero as "nothing," which sounded right, but that what they really thought was that zero was not a number, which was incorrect. Teachers also have to be able to detect that students do understand things even when they say things in ways that are not easy for adults to comprehend.

Ball and her colleagues have shown that teachers' ability to help students understand and succeed with math depends on their ability to hear and understand what students are thinking and to explain or show ideas in ways that are accessible to the students. This, in turn, depends on the teachers' own understanding. The sort of mathematical understanding needed by effective teachers is more than being able to get the right answers themselves.

The researchers developed questionnaires that test for this kind of mathematical knowledge needed for teaching. Teachers who scored higher on those questions, in turn, taught mathematics more skillfully, producing better gains in student achievement.

"Comparing a teacher who achieved an average score on our measure of teacher knowledge to a teacher who was in the top quartile, the students of the above-average teacher showed gains in their scores that were equivalent to that of an extra two to three weeks of instruction," the researchers said.

Teachers' scores on these mathematical knowledge for teaching questions had significant effects on students' learning even when compared with the effects of socioeconomic factors on school achievement.

The researchers point out that schools in poor communities are often most likely to have teachers who are under-qualified. The research demonstrated the important contribution that a knowledgeable teacher can have on students' mathematics learning.

"Our finding indicates that, while teachers' mathematical knowledge would not by itself overcome the existing achievement gap, it could prevent that that gap from growing," Ball said.